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Theorem erdszelem10 27103
Description: Lemma for erdsze 27105. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdszelem.i  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.j  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.t  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
erdszelem.r  |-  ( ph  ->  R  e.  NN )
erdszelem.s  |-  ( ph  ->  S  e.  NN )
erdszelem.m  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
Assertion
Ref Expression
erdszelem10  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
Distinct variable groups:    x, y    m, n, x, y, F   
n, I, x, y   
n, J, x, y    R, m, x, y    m, N, n, x, y    ph, m, n, x, y    S, m, x, y    T, m
Allowed substitution hints:    R( n)    S( n)    T( x, y, n)    I( m)    J( m)

Proof of Theorem erdszelem10
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 fzfi 11809 . . . . . . . 8  |-  ( 1 ... ( R  - 
1 ) )  e. 
Fin
2 fzfi 11809 . . . . . . . 8  |-  ( 1 ... ( S  - 
1 ) )  e. 
Fin
3 xpfi 7598 . . . . . . . 8  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin )
41, 2, 3mp2an 672 . . . . . . 7  |-  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  e. 
Fin
5 ssdomg 7370 . . . . . . 7  |-  ( ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  e.  Fin  ->  ( ran  T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  ran  T  ~<_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
64, 5ax-mp 5 . . . . . 6  |-  ( ran 
T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  ran  T  ~<_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
7 domnsym 7452 . . . . . 6  |-  ( ran 
T  ~<_  ( ( 1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
86, 7syl 16 . . . . 5  |-  ( ran 
T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
9 erdszelem.m . . . . . . . 8  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
10 hashxp 12211 . . . . . . . . . 10  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( (
# `  ( 1 ... ( R  -  1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) ) )
111, 2, 10mp2an 672 . . . . . . . . 9  |-  ( # `  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )  =  ( (
# `  ( 1 ... ( R  -  1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )
12 erdszelem.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
13 nnm1nn0 10636 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  ( R  -  1 )  e.  NN0 )
14 hashfz1 12132 . . . . . . . . . . 11  |-  ( ( R  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( R  -  1 ) ) )  =  ( R  -  1 ) )
1512, 13, 143syl 20 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( R  - 
1 ) ) )  =  ( R  - 
1 ) )
16 erdszelem.s . . . . . . . . . . 11  |-  ( ph  ->  S  e.  NN )
17 nnm1nn0 10636 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
18 hashfz1 12132 . . . . . . . . . . 11  |-  ( ( S  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( S  -  1 ) ) )  =  ( S  -  1 ) )
1916, 17, 183syl 20 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( S  - 
1 ) ) )  =  ( S  - 
1 ) )
2015, 19oveq12d 6124 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  (
1 ... ( R  - 
1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
2111, 20syl5eq 2487 . . . . . . . 8  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
22 erdsze.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN )
2322nnnn0d 10651 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
24 hashfz1 12132 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2523, 24syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
269, 21, 253brtr4d 4337 . . . . . . 7  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) ) )
27 fzfid 11810 . . . . . . . 8  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
28 hashsdom 12159 . . . . . . . 8  |-  ( ( ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin  /\  (
1 ... N )  e. 
Fin )  ->  (
( # `  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) )  <  ( # `  (
1 ... N ) )  <-> 
( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
) ) )
294, 27, 28sylancr 663 . . . . . . 7  |-  ( ph  ->  ( ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) )  <->  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ~<  (
1 ... N ) ) )
3026, 29mpbid 210 . . . . . 6  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
) )
31 erdsze.f . . . . . . . 8  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
32 erdszelem.i . . . . . . . 8  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
33 erdszelem.j . . . . . . . 8  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
34 erdszelem.t . . . . . . . 8  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
3522, 31, 32, 33, 34erdszelem9 27102 . . . . . . 7  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
36 f1f1orn 5667 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T : ( 1 ... N ) -1-1-onto-> ran  T )
37 ovex 6131 . . . . . . . 8  |-  ( 1 ... N )  e. 
_V
3837f1oen 7345 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-onto-> ran  T  ->  (
1 ... N )  ~~  ran  T )
3935, 36, 383syl 20 . . . . . 6  |-  ( ph  ->  ( 1 ... N
)  ~~  ran  T )
40 sdomentr 7460 . . . . . 6  |-  ( ( ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
)  /\  ( 1 ... N )  ~~  ran  T )  ->  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
4130, 39, 40syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
428, 41nsyl3 119 . . . 4  |-  ( ph  ->  -.  ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
43 nss 3429 . . . . 5  |-  ( -. 
ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  E. s
( s  e.  ran  T  /\  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
44 df-rex 2736 . . . . 5  |-  ( E. s  e.  ran  T  -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  E. s ( s  e. 
ran  T  /\  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
4543, 44bitr4i 252 . . . 4  |-  ( -. 
ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
4642, 45sylib 196 . . 3  |-  ( ph  ->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
47 f1fn 5622 . . . 4  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T  Fn  ( 1 ... N ) )
48 eleq1 2503 . . . . . 6  |-  ( s  =  ( T `  m )  ->  (
s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) ) )
4948notbid 294 . . . . 5  |-  ( s  =  ( T `  m )  ->  ( -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5049rexrn 5860 . . . 4  |-  ( T  Fn  ( 1 ... N )  ->  ( E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N )  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5135, 47, 503syl 20 . . 3  |-  ( ph  ->  ( E. s  e. 
ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N )  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5246, 51mpbid 210 . 2  |-  ( ph  ->  E. m  e.  ( 1 ... N )  -.  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
53 fveq2 5706 . . . . . . . . . 10  |-  ( n  =  m  ->  (
I `  n )  =  ( I `  m ) )
54 fveq2 5706 . . . . . . . . . 10  |-  ( n  =  m  ->  ( J `  n )  =  ( J `  m ) )
5553, 54opeq12d 4082 . . . . . . . . 9  |-  ( n  =  m  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  m ) ,  ( J `  m ) >. )
56 opex 4571 . . . . . . . . 9  |-  <. (
I `  m ) ,  ( J `  m ) >.  e.  _V
5755, 34, 56fvmpt 5789 . . . . . . . 8  |-  ( m  e.  ( 1 ... N )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5857adantl 466 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5958eleq1d 2509 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  <. ( I `
 m ) ,  ( J `  m
) >.  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
60 opelxp 4884 . . . . . 6  |-  ( <.
( I `  m
) ,  ( J `
 m ) >.  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <-> 
( ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
6159, 60syl6bb 261 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `
 m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6261notbid 294 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  -.  (
( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
63 ianor 488 . . . 4  |-  ( -.  ( ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) )  <->  ( -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
6462, 63syl6bb 261 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( -.  ( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  \/ 
-.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6564rexbidva 2747 . 2  |-  ( ph  ->  ( E. m  e.  ( 1 ... N
)  -.  ( T `
 m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N ) ( -.  ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `
 m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6652, 65mpbid 210 1  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2731   {crab 2734    C_ wss 3343   ~Pcpw 3875   <.cop 3898   class class class wbr 4307    e. cmpt 4365    X. cxp 4853   `'ccnv 4854   ran crn 4856    |` cres 4857   "cima 4858    Fn wfn 5428   -1-1->wf1 5430   -1-1-onto->wf1o 5432   ` cfv 5433    Isom wiso 5434  (class class class)co 6106    ~~ cen 7322    ~<_ cdom 7323    ~< csdm 7324   Fincfn 7325   supcsup 7705   RRcr 9296   1c1 9298    x. cmul 9302    < clt 9433    - cmin 9610   NNcn 10337   NN0cn0 10594   ...cfz 11452   #chash 12118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-n0 10595  df-z 10662  df-uz 10877  df-fz 11453  df-hash 12119
This theorem is referenced by:  erdszelem11  27104
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